Statistics Seminar: Samuel Mueller -- Smooth extreme value estimation for densities with log-concave tails

Samuel Mueller School of Mathematics and Statistics University of Sydney 

Location: Carslaw 173 

Time: 2pm Friday, April 24, 2009 

Title: Smooth extreme value estimation for densities with log-concave tails 

Abstract: It is shown that (1) both parametric distribution functions appearing in
extreme value theory -- the generalized extreme value distribution and the generalized
Pareto distribution -- have log-concave densities if the extreme value index is in
[-1,0] and (2) that all distribution functions with log-concave density belong to the
max--domain of attraction of the generalized extreme value distribution.  Many extreme
value tail-index estimators, such as (generalized) Pickands, are functions of the upper
order statistics.  Therefore, one can replace these order statistics by their
corresponding smoothed quantiles from the distribution function that is based on the
estimated log-concave density.  This leads to smooth quantile and tail index
estimators.  These new estimators are particularly useful in small samples.  Acting as a
smoother of the empirical distribution function, the log-concave distribution function
estimator reduces estimation variability to a much greater extent than it introduces
bias.  As a consequence, Monte Carlo simulations demonstrate that the smoothed version
of the estimators are superior to their non-smoothed counterparts, in terms of mean
squared error.  The R package smoothtail is presented, which provides functions to
calculate the smoothed estimators.  

References [1] Mueller, S., Rufibach, K.  (2007).  Software smoothtail: Smooth
estimation of GPD shape parameter.  R package version 1.1.2.  [2] Mueller, S., Rufibach,
K.  (2008).  On the Max-Domain of Attraction of Distributions with Log-Concave
Densities.  Statistics & Probability Letters78:1440-1444.  [3] Mueller, S., Rufibach,
K.  (2009 to appear).  Smooth Tail Index Estimators.  Journal of Statistical Computation
and Simulation, 13 pages.